It's a standard theorem that the number of ways to write a positive integer N as the sum of *two* squares is given by four times the difference between its number of divisors which are congruent to 1 mod 4 and its number of divisors which are congruent to 3 mod 4. Alternatively, there are no such representations if the prime factorization of N contains any prime of form 4k+3 an odd number of times. If the prime factorization of N contains all such primes an even number of times, then we have

r_{2}(N) = 4(b_{1} + 1)(b_{2} + 1)...(b_{r}+1)

where b_{1}, ..., b_{r} are the exponents of the primes congruent to 1 mod 4 in the factorization of N.

For example, 325 = 5^{2} × 13 can be written in 4(2+1)(1+1) = 24 ways as a sum of squares. These are 18^{2} + 1^{2}, 17^{2} + 6^{2}, 15^{2} + 10^{2}, and the representations obtained from these by changing signs and/or permuting.

Is there an analogous formula in the three-square case? I know that an integer can be written as the sum of three squares if and only if it is not of the form 4^{m}(8n+7). There is a simple argument that shows that the number of ways to write all integers up to N as a sum of three squares is asymptotically 4πN^{3/2}/3 -- representations of an integer less than N as a sum of three squares can be identified with points in the ball in **R**^{3} centered at the origin with radius N^{1/2}. Differentiating, a "typical" integer near N should have about 2πN^{1/2} representations as a sum of three squares. From playing around with some data it looks like

lim_{n → ∞} #{ k ≤ n and r_{3}(k)/k^{1/2} ≤ x} / n

might be a nonzero constant. That is, for each positive real x, the probability that a random integer k can be written in no more than x k^{1/2} ways approaches some constant in the open interval (0, 1) as k → ∞.

One way to prove this (if it is in fact true) would be if there were some formula for r_{3}(k), in terms of the prime factorization, which is why I'm curious.

(I apologize if this is something that is well-known to number theorists, although I'd appreciate a pointer if it is. I am *not* a number theorist, I just play around with this sort of thing every so often and generate amusing conjectures.)